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   <h4 class="subsectionHead"><span class="titlemark">2.14   </span> <a 
 id="x19-650002.14"></a>Geometric nonlinear analysis</h4>
<!--l. 1536--><p class="noindent" >To take int account geometric nonlinearity for a specific element, the keyword <span 
class="cmtt-10">nlgeo </span>must be specified. The <span 
class="cmtt-10">nlgeo</span>
parameter defines which formulation of the momentum balance is solved and what deformation measure that is
computed and sent to the consitiutive models (see Table&#x00A0;<a 
href="#x19-6500549">49<!--tex4ht:ref: strain_tensor_table --></a>). If <span 
class="cmtt-10">nlgeo</span>=1, then the momentum balance is set up in the
reference configuration in terms of the First Piola-Kirchhoff stress tensor <span 
class="cmbx-10">P </span>and the deformation tensor <span 
class="cmbx-10">F </span>as energy
conjugates. This is also refered to a Total Lagrangian formulation. The balance equation in weak form
reads
   <table 
class="equation"><tr><td><a 
 id="x19-65001r6"></a>
   <center class="math-display" >
<img 
src="elementlibmanual8x.png" alt="&int;            &int;           &int;
  &delta;F : Pd &Omega; =  &delta;x&sdot;t  d&Gamma; +    &delta;x&sdot;b  d&Omega;
 &Omega;            &Gamma;    P      &part;&Omega;     P
" class="math-display" ></center></td><td class="equation-label">(6)</td></tr></table>
<!--l. 1541--><p class="nopar" >
This equation can be rewritten in terms of the displacement <span 
class="cmbx-10">u </span>and the displacement gradient <span 
class="cmbx-10">H</span>
   <table 
class="equation"><tr><td><a 
 id="x19-65002r7"></a>
   <center class="math-display" >
<img 
src="elementlibmanual9x.png" alt="&int;             &int;           &int;
   &delta;H : Pd &Omega; =  &delta;u &sdot;tP d&Gamma; +    &delta;u&sdot;bP d&Omega;
 &Omega;             &Gamma;           &part;&Omega;
" class="math-display" ></center></td><td class="equation-label">(7)</td></tr></table>
<!--l. 1547--><p class="nopar" >
This equation is nearly identical to the one for small strains except that another stress measure is used and we have
the virtual displacement gradient instead of the virtual strains.
<!--l. 1551--><p class="indent" >   The corresponding FE-formulation is obtained as
   <table 
class="equation"><tr><td><a 
 id="x19-65003r8"></a>
   <center class="math-display" >
<img 
src="elementlibmanual10x.png" alt="&int;             &int;            &int;
   BTH &sdot;Pd &Omega; =   NT &sdot;tPd&Gamma; +    &delta;NT  &sdot;bPd&Omega;
 &Omega;             &Gamma;            &part;&Omega;
" class="math-display" ></center></td><td class="equation-label">(8)</td></tr></table>
<!--l. 1555--><p class="nopar" >
with the tangent stiffness
   <table 
class="equation"><tr><td><a 
 id="x19-65004r9"></a>
   <center class="math-display" >
<img 
src="elementlibmanual11x.png" alt="      &int;      &part;P
KT  =   BTH &sdot;---&sdot;BHd &Omega;
       &Omega;     &part;F
" class="math-display" ></center></td><td class="equation-label">(9)</td></tr></table>
<!--l. 1559--><p class="nopar" >
Thus, for an element to support large deformations (in addition to small deformation) it needs only to implement the
<span 
class="cmbx-10">B</span><sub><span 
class="cmmi-7">H</span></sub> matrix. Similar to the regular <span 
class="cmbx-10">B </span>matrix, which gives the strains in Voigt form when multiplied with the solution
vector <span 
class="cmbx-10">a</span>, <span 
class="cmbx-10">B</span><sub><span 
class="cmmi-7">H</span></sub> should give the displcement gradient in Voigt form with 9 components for a full 3D state.
<div class="table">
                                                                                           
                                                                                           
<!--l. 1565--><p class="indent" >   <a 
 id="x19-6500549"></a><hr class="float"><div class="float" 
>
                                                                                           
                                                                                           
<div class="tabular"> <table id="TBL-51" class="tabular" 
cellspacing="0" cellpadding="0"  
><colgroup id="TBL-51-1g"><col 
id="TBL-51-1"><col 
id="TBL-51-2"><col 
id="TBL-51-3"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-51-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-51-1-1"  
class="td11">nlgeo      </td><td  style="white-space:nowrap; text-align:left;" id="TBL-51-1-2"  
class="td11">strain tensor                      </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-51-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-51-2-1"  
class="td11">0 (default)</td><td  style="white-space:nowrap; text-align:left;" id="TBL-51-2-2"  
class="td11">Small-strain tensor              </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-51-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-51-3-1"  
class="td11">1            </td><td  style="white-space:nowrap; text-align:left;" id="TBL-51-3-2"  
class="td11">Green-Lagrange strain tensor</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-51-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-51-4-1"  
class="td11">2            </td><td  style="white-space:nowrap; text-align:left;" id="TBL-51-4-2"  
class="td11">Deformation gradient          </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-51-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-51-5-1"  
class="td11">         </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;49: </span><span  
class="content">Nonlinear geometry modes</span></div><!--tex4ht:label?: x19-6500549 -->
                                                                                           
                                                                                           
   </div><hr class="endfloat" />
   </div>
                                                                                           
                                                                                           
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